How can I thicken the walls of my model for a more sturdy 3D printing?
The quick solution to make a Blender model of this is to export it as an stl then import it in Blender. Blender can read stl files. That's the quick, though often dirty, solution. It works most of the time, though you often have to check the model and redo some stuff.

I believe that the inner geometry of the piece is unnecessarily complex, thus why it generates so much problem. The gif, while fast, seems to show some problem arising when the model is sliced and it seems that there are missing layers and holes when I compare the end of the gif to the picture of the 3D model you provided.

That being said, my main concern, if I interpret your gif right, is the insane movements of the printer head! If I'm not mistaken, the thin red lines on the gif must represent the movements of the printhead, and they're all over the place! With such a geometric pattern inside, I would expect the printhead to follow the lines, but yours seems to jump all around.

That would also explain why your print fails: the printhead has to do too many retractions.

I would suggest to simply the inner geometry, and you can thicken the walls of an object using the 'Solidify' modifiers in Blender.

Although, I think it would be best to just rebuild it from scratch in Blender and make the walls thicker from the start.